Subgroups — C₂×C₅₃×#1646 #27226026

Order 13568.

1042 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#555	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#487	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅₃×#474	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅₃×#461	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1646	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₅₃×#333	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅₃×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅₃×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×SD₁₆×C₅₃	—	2	1	no	—
10	C₅₃×#555	—	2	2	no	—
11	C₅₃×#555	—	2	2	no	—
12	C₅₃×#555	—	2	2	no	—
13	C₅₃×#555	C₂²	1	4	yes	—
14	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#124×C₅₃	—	2	2	no	—
17	C₂×#121×C₅₃	—	2	1	no	—
18	C₂×#121×C₅₃	—	2	1	no	—
19	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×#117×C₅₃	—	2	1	no	—
22	C₂×#117×C₅₃	—	2	1	no	—
23	C₅₃×#487	—	2	2	no	—
24	C₅₃×#487	C₂²	1	4	yes	complemented
25	C₅₃×#474	—	2	2	no	—
26	C₅₃×#474	C₂²	1	4	yes	complemented
27	C₅₃×#461	—	2	2	no	—
28	C₅₃×#461	C₂²	1	4	yes	complemented
29	C₂×#107×C₅₃	—	2	1	no	—
30	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
32	C₂×#103×C₅₃	—	2	2	no	—
33	C₂×#103×C₅₃	—	2	1	no	—
34	C₂×#102×C₅₃	—	2	1	no	—
35	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
36	C₅₃×#333	C₂²	1	4	yes	complemented
37	C₅₃×D₄⋊C₈	C₂²	1	4	yes	complemented
38	C₅₃×D₄⋊C₈	C₂²	1	4	yes	complemented
39	C₂²×Q₈×C₅₃	—	2	1	no	—
40	C₂²×Q₈×C₅₃	D₄	1	2	yes	—
41	C₂²×D₄×C₅₃	—	2	2	no	—
42	C₂²×D₄×C₅₃	D₄	1	1	yes	characteristic
43	C₂²×D₄×C₅₃	D₄	1	1	yes	characteristic
44	C₂³×C₄×C₅₃	—	2	1	no	—
45	C₂³×C₄×C₅₃	—	2	1	no	—
46	C₂×SD₁₆×C₅₃	—	2	4	no	—
47	C₂×SD₁₆×C₅₃	—	4	4	no	—
48	C₂²×C₈×C₅₃	—	2	1	no	—
49	C₂²×C₈×C₅₃	—	2	1	no	—
50	C₂²×C₈×C₅₃	—	2	1	no	—
51	C₄⋊Q₈×C₅₃	—	2	1	no	—
52	C₄⋊Q₈×C₅₃	—	2	1	no	—
53	C₄⋊Q₈×C₅₃	—	2	1	no	—
54	C₄⋊Q₈×C₅₃	—	2	1	no	—
55	C₄⋊Q₈×C₅₃	—	2	1	no	—
56	C₄⋊Q₈×C₅₃	D₄	1	2	yes	—
57	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
58	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
59	#124×C₅₃	—	2	8	no	—
60	#124×C₅₃	—	2	8	no	—
61	#121×C₅₃	—	2	4	no	—
62	#121×C₅₃	—	2	4	no	—
63	#121×C₅₃	—	2	4	no	—
64	#121×C₅₃	—	2	4	no	—
65	C₄×D₄×C₅₃	—	2	1	no	—
66	C₄×D₄×C₅₃	—	2	1	no	—
67	C₄×D₄×C₅₃	—	2	1	no	—
68	C₄×D₄×C₅₃	—	2	1	no	—
69	C₄×D₄×C₅₃	D₄	1	2	yes	—
70	C₄×D₄×C₅₃	D₄	1	2	yes	—
71	C₄×D₄×C₅₃	C₂³	1	2	yes	—
72	C₄×D₄×C₅₃	C₂³	1	2	yes	—
73	#117×C₅₃	—	2	4	no	—
74	#117×C₅₃	—	2	4	no	—
75	#117×C₅₃	—	2	4	no	—
76	#117×C₅₃	—	2	4	no	—
77	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
78	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
79	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
80	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
81	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
83	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
84	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
85	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
86	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
87	C₂×C₄²×C₅₃	—	2	2	no	—
88	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
89	#107×C₅₃	—	2	4	no	—
90	C₄⋊C₈×C₅₃	—	2	1	no	—
91	C₄⋊C₈×C₅₃	—	2	1	no	—
92	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
93	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
94	#103×C₅₃	—	2	4	no	—
95	#103×C₅₃	—	2	8	no	—
96	#102×C₅₃	—	2	4	no	—
97	C₄×C₈×C₅₃	—	2	1	no	—
98	C₄×C₈×C₅₃	C₂³	1	2	yes	—
99	C₂⁴×C₅₃	—	2	1	no	—
100	C₂⁴×C₅₃	—	2	1	no	—
101	C₂×Q₈×C₅₃	—	2	2	no	—
102	C₂×Q₈×C₅₃	—	2	2	no	—
103	C₂×Q₈×C₅₃	—	4	4	no	—
104	C₂×Q₈×C₅₃	—	4	1	no	—
105	C₂×Q₈×C₅₃	—	4	2	no	—
106	C₂×Q₈×C₅₃	—	4	2	no	—
107	C₂×Q₈×C₅₃	C₂×D₄	1	4	yes	—
108	C₂×D₄×C₅₃	—	2	8	no	—
109	C₂×D₄×C₅₃	—	2	1	no	—
110	C₂×D₄×C₅₃	—	2	1	no	—
111	C₂×D₄×C₅₃	—	2	2	no	—
112	C₂×D₄×C₅₃	—	2	2	no	—
113	C₂×D₄×C₅₃	—	2	2	no	—
114	C₂×D₄×C₅₃	—	2	2	no	—
115	C₂×D₄×C₅₃	—	8	4	no	—
116	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
117	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
118	C₂²×C₄×C₅₃	—	2	1	no	—
119	C₂²×C₄×C₅₃	—	2	1	no	—
120	C₂²×C₄×C₅₃	—	2	2	no	—
121	C₂²×C₄×C₅₃	—	2	2	no	—
122	C₂²×C₄×C₅₃	—	2	2	no	—
123	C₂²×C₄×C₅₃	—	2	2	no	—
124	C₂²×C₄×C₅₃	—	2	2	no	—
125	C₂²×C₄×C₅₃	—	4	2	no	—
126	C₂²×C₄×C₅₃	—	4	2	no	—
127	C₂²×C₄×C₅₃	—	4	2	no	—
128	C₂²×C₄×C₅₃	—	4	2	no	—
129	C₂²×C₄×C₅₃	—	4	1	no	—
130	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
133	SD₁₆×C₅₃	—	4	8	no	—
134	C₂×C₈×C₅₃	—	2	2	no	—
135	C₂×C₈×C₅₃	—	2	2	no	—
136	C₂×C₈×C₅₃	—	2	2	no	—
137	C₂×C₈×C₅₃	—	4	2	no	—
138	C₂×C₈×C₅₃	—	4	2	no	—
139	C₂×C₈×C₅₃	—	4	2	no	—
140	C₄⋊C₄×C₅₃	—	2	2	no	—
141	C₄⋊C₄×C₅₃	—	2	2	no	—
142	C₄⋊C₄×C₅₃	—	2	1	no	—
143	C₄⋊C₄×C₅₃	—	2	1	no	—
144	C₄⋊C₄×C₅₃	—	2	2	no	—
145	C₄⋊C₄×C₅₃	—	2	2	no	—
146	C₄⋊C₄×C₅₃	—	2	2	no	—
147	C₄⋊C₄×C₅₃	—	2	2	no	—
148	C₄⋊C₄×C₅₃	—	4	1	no	—
149	C₄⋊C₄×C₅₃	—	4	1	no	—
150	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
151	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
152	C₂²⋊C₄×C₅₃	—	2	8	no	—
153	C₂²⋊C₄×C₅₃	—	2	8	no	—
154	C₂²⋊C₄×C₅₃	—	2	2	no	—
155	C₂²⋊C₄×C₅₃	—	2	2	no	—
156	C₂²⋊C₄×C₅₃	—	2	2	no	—
157	C₂²⋊C₄×C₅₃	—	2	2	no	—
158	C₄²×C₅₃	—	2	8	no	—
159	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
160	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
161	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
162	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
163	C₂³×C₅₃	—	2	2	no	—
164	C₂³×C₅₃	—	2	2	no	—
165	C₂³×C₅₃	—	4	2	no	—
166	C₂³×C₅₃	—	4	2	no	—
167	C₂³×C₅₃	—	4	2	no	—
168	C₂³×C₅₃	—	4	2	no	—
169	C₂³×C₅₃	—	4	2	no	—
170	C₂³×C₅₃	—	4	2	no	—
171	C₂³×C₅₃	#120	1	1	yes	characteristic
172	Q₈×C₅₃	—	4	4	no	—
173	Q₈×C₅₃	—	4	2	no	—
174	Q₈×C₅₃	—	4	2	no	—
175	Q₈×C₅₃	—	8	2	no	—
176	Q₈×C₅₃	—	8	2	no	—
177	D₄×C₅₃	—	2	2	no	—
178	D₄×C₅₃	—	2	2	no	—
179	D₄×C₅₃	—	2	2	no	—
180	D₄×C₅₃	—	2	2	no	—
181	D₄×C₅₃	—	4	2	no	—
182	D₄×C₅₃	—	4	2	no	—
183	D₄×C₅₃	—	8	8	no	—
184	C₂×C₄×C₅₃	—	2	2	no	—
185	C₂×C₄×C₅₃	—	2	2	no	—
186	C₂×C₄×C₅₃	—	2	2	no	—
187	C₂×C₄×C₅₃	—	2	4	no	—
188	C₂×C₄×C₅₃	—	4	2	no	—
189	C₂×C₄×C₅₃	—	4	2	no	—
190	C₂×C₄×C₅₃	—	4	4	no	—
191	C₂×C₄×C₅₃	—	4	2	no	—
192	C₂×C₄×C₅₃	—	4	2	no	—
193	C₂×C₄×C₅₃	—	4	2	no	—
194	C₂×C₄×C₅₃	—	4	2	no	—
195	C₂×C₄×C₅₃	—	4	4	no	—
196	C₂×C₄×C₅₃	—	4	4	no	—
197	C₂×C₄×C₅₃	—	4	2	no	—
198	C₂×C₄×C₅₃	—	8	2	no	—
199	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
200	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
201	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
202	C₂×C₄×C₅₃	#135	1	2	yes	—
203	C₂×C₄×C₅₃	#135	1	2	yes	—
204	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
205	C₂×C₄×C₅₃	#137	1	2	yes	—
206	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
207	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
208	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
209	C₈×C₅₃	—	4	2	no	—
210	C₈×C₅₃	—	4	2	no	—
211	C₈×C₅₃	—	4	2	no	—
212	C₂×#1646	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
213	C₂²×C₅₃	—	4	2	no	—
214	C₂²×C₅₃	—	4	2	no	—
215	C₂²×C₅₃	—	4	2	no	—
216	C₂²×C₅₃	—	4	2	no	—
217	C₂²×C₅₃	—	4	2	no	—
218	C₂²×C₅₃	—	4	2	no	—
219	C₂²×C₅₃	—	8	4	no	—
220	C₂²×C₅₃	—	8	4	no	—
221	C₂²×C₅₃	#449	1	2	yes	—
222	C₂²×C₅₃	#452	1	2	yes	—
223	C₂²×C₅₃	#455	1	2	yes	—
224	C₂²×C₅₃	C₂×#120	1	1	yes	characteristic
225	C₄×C₅₃	—	2	1	no	—
226	C₄×C₅₃	—	2	1	no	—
227	C₄×C₅₃	—	4	2	no	—
228	C₄×C₅₃	—	4	2	no	—
229	C₄×C₅₃	—	4	4	no	—
230	C₄×C₅₃	—	8	2	no	—
231	C₄×C₅₃	#449	1	2	yes	—
232	C₄×C₅₃	#452	1	2	yes	—
233	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
234	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
235	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
236	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
237	C₂×#555	C₂×C₅₃	1	1	yes	characteristic
238	C₂×#487	C₂×C₅₃	1	1	yes	characteristic, complemented
239	C₂×#474	C₂×C₅₃	1	1	yes	characteristic, complemented
240	C₂×#461	C₂×C₅₃	1	1	yes	characteristic, complemented
241	#1646	C₂×C₅₃	1	8	yes	direct product factor
242	C₂×#333	C₂×C₅₃	1	1	yes	characteristic, complemented
243	C₂×D₄⋊C₈	C₂×C₅₃	1	1	yes	characteristic, complemented
244	C₂×D₄⋊C₈	C₂×C₅₃	1	1	yes	characteristic, complemented
245	C₂×C₅₃	—	8	2	no	—
246	C₂×C₅₃	—	8	2	no	—
247	C₂×C₅₃	#1646	1	4	yes	direct product factor
248	C₂×C₅₃	C₂×#449	1	1	yes	characteristic
249	C₂×C₅₃	C₂×#452	1	1	yes	characteristic
250	C₂×C₅₃	C₂×#455	1	1	yes	characteristic
251	C₂²×SD₁₆	—	2	1	no	—
252	#555	—	2	2	no	—
253	#555	—	2	2	no	—
254	#555	—	2	2	no	—
255	#555	C₂²×C₅₃	1	4	yes	—
256	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
257	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
258	C₂×#124	—	2	2	no	—
259	C₂×#121	—	2	1	no	—
260	C₂×#121	—	2	1	no	—
261	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
262	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
263	C₂×#117	—	2	1	no	—
264	C₂×#117	—	2	1	no	—
265	#487	—	2	2	no	—
266	#487	C₂²×C₅₃	1	4	yes	complemented
267	#474	—	2	2	no	—
268	#474	C₂²×C₅₃	1	4	yes	complemented
269	#461	—	2	2	no	—
270	#461	C₂²×C₅₃	1	4	yes	complemented
271	C₂×#107	—	2	1	no	—
272	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
273	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
274	C₂×#103	—	2	2	no	—
275	C₂×#103	—	2	1	no	—
276	C₂×#102	—	2	1	no	—
277	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
278	#333	C₂²×C₅₃	1	4	yes	complemented
279	D₄⋊C₈	C₂²×C₅₃	1	4	yes	complemented
280	D₄⋊C₈	C₂²×C₅₃	1	4	yes	complemented
281	C₅₃	C₂×#1646	1	1	yes	characteristic, direct product factor, Sylow
282	C₂²×Q₈	—	2	1	no	—
283	C₂²×Q₈	D₄×C₅₃	1	2	yes	—
284	C₂²×D₄	—	2	2	no	—
285	C₂²×D₄	D₄×C₅₃	1	1	yes	characteristic
286	C₂²×D₄	D₄×C₅₃	1	1	yes	characteristic
287	C₂³×C₄	—	2	1	no	—
288	C₂³×C₄	—	2	1	no	—
289	C₂×SD₁₆	—	2	4	no	—
290	C₂×SD₁₆	—	4	4	no	—
291	C₂²×C₈	—	2	1	no	—
292	C₂²×C₈	—	2	1	no	—
293	C₂²×C₈	—	2	1	no	—
294	C₄⋊Q₈	—	2	1	no	—
295	C₄⋊Q₈	—	2	1	no	—
296	C₄⋊Q₈	—	2	1	no	—
297	C₄⋊Q₈	—	2	1	no	—
298	C₄⋊Q₈	—	2	1	no	—
299	C₄⋊Q₈	D₄×C₅₃	1	2	yes	—
300	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
301	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
302	#124	—	2	8	no	—
303	#124	—	2	8	no	—
304	#121	—	2	4	no	—
305	#121	—	2	4	no	—
306	#121	—	2	4	no	—
307	#121	—	2	4	no	—
308	C₄×D₄	—	2	1	no	—
309	C₄×D₄	—	2	1	no	—
310	C₄×D₄	—	2	1	no	—
311	C₄×D₄	—	2	1	no	—
312	C₄×D₄	D₄×C₅₃	1	2	yes	—
313	C₄×D₄	D₄×C₅₃	1	2	yes	—
314	C₄×D₄	C₂³×C₅₃	1	2	yes	—
315	C₄×D₄	C₂³×C₅₃	1	2	yes	—
316	#117	—	2	4	no	—
317	#117	—	2	4	no	—
318	#117	—	2	4	no	—
319	#117	—	2	4	no	—
320	C₂×C₄⋊C₄	—	2	1	no	—
321	C₂×C₄⋊C₄	—	2	1	no	—
322	C₂×C₄⋊C₄	—	2	1	no	—
323	C₂×C₄⋊C₄	—	2	1	no	—
324	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
325	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
326	C₂×C₂²⋊C₄	—	2	2	no	—
327	C₂×C₂²⋊C₄	—	2	2	no	—
328	C₂×C₂²⋊C₄	—	2	1	no	—
329	C₂×C₂²⋊C₄	—	2	1	no	—
330	C₂×C₄²	—	2	2	no	—
331	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
332	#107	—	2	4	no	—
333	C₄⋊C₈	—	2	1	no	—
334	C₄⋊C₈	—	2	1	no	—
335	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
336	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
337	#103	—	2	8	no	—
338	#103	—	2	4	no	—
339	#102	—	2	4	no	—
340	C₄×C₈	—	2	1	no	—
341	C₄×C₈	C₂³×C₅₃	1	2	yes	—
342	C₂⁴	—	2	1	no	—
343	C₂⁴	—	2	1	no	—
344	C₂×Q₈	—	2	2	no	—
345	C₂×Q₈	—	2	2	no	—
346	C₂×Q₈	—	4	4	no	—
347	C₂×Q₈	—	4	1	no	—
348	C₂×Q₈	—	4	2	no	—
349	C₂×Q₈	—	4	2	no	—
350	C₂×Q₈	C₂×D₄×C₅₃	1	4	yes	—
351	C₂×D₄	—	2	8	no	—
352	C₂×D₄	—	2	1	no	—
353	C₂×D₄	—	2	1	no	—
354	C₂×D₄	—	2	2	no	—
355	C₂×D₄	—	2	2	no	—
356	C₂×D₄	—	2	2	no	—
357	C₂×D₄	—	2	2	no	—
358	C₂×D₄	—	8	4	no	—
359	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
360	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
361	C₂²×C₄	—	2	1	no	—
362	C₂²×C₄	—	2	1	no	—
363	C₂²×C₄	—	2	2	no	—
364	C₂²×C₄	—	2	2	no	—
365	C₂²×C₄	—	2	2	no	—
366	C₂²×C₄	—	2	2	no	—
367	C₂²×C₄	—	2	2	no	—
368	C₂²×C₄	—	4	2	no	—
369	C₂²×C₄	—	4	2	no	—
370	C₂²×C₄	—	4	2	no	—
371	C₂²×C₄	—	4	2	no	—
372	C₂²×C₄	—	4	1	no	—
373	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
374	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
375	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
376	SD₁₆	—	4	8	no	—
377	C₂×C₈	—	2	2	no	—
378	C₂×C₈	—	2	2	no	—
379	C₂×C₈	—	2	2	no	—
380	C₂×C₈	—	4	2	no	—
381	C₂×C₈	—	4	2	no	—
382	C₂×C₈	—	4	2	no	—
383	C₄⋊C₄	—	2	2	no	—
384	C₄⋊C₄	—	2	2	no	—
385	C₄⋊C₄	—	2	1	no	—
386	C₄⋊C₄	—	2	1	no	—
387	C₄⋊C₄	—	2	2	no	—
388	C₄⋊C₄	—	2	2	no	—
389	C₄⋊C₄	—	2	2	no	—
390	C₄⋊C₄	—	2	2	no	—
391	C₄⋊C₄	—	4	1	no	—
392	C₄⋊C₄	—	4	1	no	—
393	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
394	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
395	C₂²⋊C₄	—	2	8	no	—
396	C₂²⋊C₄	—	2	8	no	—
397	C₂²⋊C₄	—	2	2	no	—
398	C₂²⋊C₄	—	2	2	no	—
399	C₂²⋊C₄	—	2	2	no	—
400	C₂²⋊C₄	—	2	2	no	—
401	C₄²	—	2	8	no	—
402	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
403	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
404	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
405	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
406	C₂³	—	2	2	no	—
407	C₂³	—	2	2	no	—
408	C₂³	—	4	2	no	—
409	C₂³	—	4	2	no	—
410	C₂³	—	4	2	no	—
411	C₂³	—	4	2	no	—
412	C₂³	—	4	2	no	—
413	C₂³	—	4	2	no	—
414	C₂³	#120×C₅₃	1	1	yes	characteristic
415	Q₈	—	4	4	no	—
416	Q₈	—	4	2	no	—
417	Q₈	—	4	2	no	—
418	Q₈	—	8	2	no	—
419	Q₈	—	8	2	no	—
420	D₄	—	2	2	no	—
421	D₄	—	2	2	no	—
422	D₄	—	2	2	no	—
423	D₄	—	2	2	no	—
424	D₄	—	4	2	no	—
425	D₄	—	4	2	no	—
426	D₄	—	8	8	no	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	4	no	—
431	C₂×C₄	—	4	2	no	—
432	C₂×C₄	—	4	2	no	—
433	C₂×C₄	—	4	4	no	—
434	C₂×C₄	—	4	2	no	—
435	C₂×C₄	—	4	2	no	—
436	C₂×C₄	—	4	2	no	—
437	C₂×C₄	—	4	2	no	—
438	C₂×C₄	—	4	4	no	—
439	C₂×C₄	—	4	4	no	—
440	C₂×C₄	—	4	2	no	—
441	C₂×C₄	—	8	2	no	—
442	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
443	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
444	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
445	C₂×C₄	#135×C₅₃	1	2	yes	—
446	C₂×C₄	#135×C₅₃	1	2	yes	—
447	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
448	C₂×C₄	#137×C₅₃	1	2	yes	—
449	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
450	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
451	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
452	C₈	—	4	2	no	—
453	C₈	—	4	2	no	—
454	C₈	—	4	2	no	—
455	C₂²	—	4	2	no	—
456	C₂²	—	4	2	no	—
457	C₂²	—	4	2	no	—
458	C₂²	—	4	2	no	—
459	C₂²	—	4	2	no	—
460	C₂²	—	4	2	no	—
461	C₂²	—	8	4	no	—
462	C₂²	—	8	4	no	—
463	C₂²	C₅₃×#449	1	2	yes	—
464	C₂²	C₅₃×#452	1	2	yes	—
465	C₂²	C₅₃×#455	1	2	yes	—
466	C₂²	C₂×#120×C₅₃	1	1	yes	characteristic
467	C₄	—	2	1	no	—
468	C₄	—	2	1	no	—
469	C₄	—	4	2	no	—
470	C₄	—	4	2	no	—
471	C₄	—	4	4	no	—
472	C₄	—	8	2	no	—
473	C₄	C₅₃×#449	1	2	yes	—
474	C₄	C₅₃×#452	1	2	yes	—
475	C₄	C₂×#135×C₅₃	1	1	yes	characteristic
476	C₄	C₂×#135×C₅₃	1	1	yes	characteristic
477	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
478	C₄	C₂×#137×C₅₃	1	1	yes	characteristic
479	C₂	—	8	2	no	—
480	C₂	—	8	2	no	—
481	C₂	C₅₃×#1646	1	4	yes	direct product factor
482	C₂	C₂×C₅₃×#449	1	1	yes	characteristic
483	C₂	C₂×C₅₃×#452	1	1	yes	characteristic
484	C₂	C₂×C₅₃×#455	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
prime radical	C₂×C₅₃×#555	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂×C₅₃×#487	C₂	semidirect
3. C₂×C₅₃×#474	C₂	semidirect
4. C₂×C₅₃×#461	C₂	semidirect
5. C₅₃×#1646	C₂	direct (N × H)
6. C₂×C₅₃×#333	C₂	semidirect
7. C₂×C₅₃×D₄⋊C₈	C₂	semidirect
8. C₂×C₅₃×D₄⋊C₈	C₂	semidirect
24. C₅₃×#487	C₂²	semidirect
26. C₅₃×#474	C₂²	semidirect
28. C₅₃×#461	C₂²	semidirect
36. C₅₃×#333	C₂²	semidirect
37. C₅₃×D₄⋊C₈	C₂²	semidirect
38. C₅₃×D₄⋊C₈	C₂²	semidirect
212. C₂×#1646	C₅₃	direct (N × H)
238. C₂×#487	C₂×C₅₃	semidirect
239. C₂×#474	C₂×C₅₃	semidirect
240. C₂×#461	C₂×C₅₃	semidirect
241. #1646	C₂×C₅₃	direct (N × H)
242. C₂×#333	C₂×C₅₃	semidirect
243. C₂×D₄⋊C₈	C₂×C₅₃	semidirect
244. C₂×D₄⋊C₈	C₂×C₅₃	semidirect
247. C₂×C₅₃	#1646	direct (N × H)
266. #487	C₂²×C₅₃	semidirect
268. #474	C₂²×C₅₃	semidirect
270. #461	C₂²×C₅₃	semidirect
278. #333	C₂²×C₅₃	semidirect
279. D₄⋊C₈	C₂²×C₅₃	semidirect
280. D₄⋊C₈	C₂²×C₅₃	semidirect
281. C₅₃	C₂×#1646	direct (N × H)
481. C₂	C₅₃×#1646	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₅₃×#555	2
C₂×C₅₃×#487	2
C₂×C₅₃×#474	2
C₂×C₅₃×#461	2
C₅₃×#1646	2
C₂×C₅₃×#333	2
C₂×C₅₃×D₄⋊C₈	2
C₂×#1646	53

Subgroups — C2×C53×#1646 #27226026

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#1646 #27226026